A Generalized Richardson Number Diagnostic for Turbulence in the Free Atmosphere

Abstract A new Richardson number formulation, Ri new , is introduced to improve the diagnosis of turbulence in the stratified free atmosphere. The formulation is derived from the turbulent kinetic energy budget and accounts for both vertical wind shear and horizontal shear (deformation and divergence), weighted by the ratio of horizontal to vertical eddy viscosities ( K mh / K mυ ). This extends the classical Richardson number Ri old , which accounts only for vertical shear. The diagnostics Ri new , Ri old , and the widely used turbulence index 1 (TI1), computed from ERA5 reanalysis, are evaluated using more than 247 million automated turbulence reports from commercial aircraft (2017–24). Across various turbulence intensity thresholds, Ri new consistently outperforms the other diagnostics, resulting in higher area under the receiver operating characteristic (ROC) curve (AUC) values and improved probability of detection at operationally relevant false alarm rates. The highest skill is obtained for K mh / K mυ ≈ 5000. Seasonal and regional evaluations indicate that the added value of Ri new is largest where turbulence generation involves both vertical and horizontal shear, such as over the contiguous United States and during summer. The Ri new remains the best-performing diagnostic in all regions and seasons. Spatial case studies show that Ri new identifies 83%–98% of observed moderate-or-greater turbulence events compared with 54%–85% for Ri old . This substantial improvement in detection comes with a much smaller increase in false alarms, confirming that Ri new provides a more physically realistic representation of turbulence-prone regions. These results demonstrate that incorporating horizontal wind shear into the Richardson number yields a physically consistent and statistically robust improvement in turbulence diagnostics, with relevance for research and operational applications. Significance Statement Turbulence at cruising altitudes is a major aviation hazard and is expected to increase as the climate warms. The Richardson number is a classical measure of the balance between turbulence suppression by atmospheric stability and generation by vertical wind shear. Here, we introduce an extension that also accounts for horizontal wind shear, which can be important near jet streams. Using more than 247 million aircraft turbulence reports, we show that this formulation consistently improves turbulence detection compared to the traditional Richardson number and a widely used turbulence index. The improvement is robust across turbulence intensities, seasons, and regions. These findings provide a more physically consistent approach to diagnosing turbulence and may support advances in aviation forecasting and climate impact assessments.